Inverse Laplace transform

Inverse Laplace transform

Contents

Mellin's inverse formula

An integral formula for the inverse Laplace transform, called the Bromwich integral, the Fourier–Mellin integral, and Mellin's inverse formula, is given by the line integral:

\mathcal{L}^{-1} \{F(s)\} = f(t) = \frac{1}{2\pi i}\lim_{T\to\infty}\int_{\gamma-iT}^{\gamma+iT}e^{st}F(s)\,ds,

where the integration is done along the vertical line Re(s) = γ in the complex plane such that γ is greater than the real part of all singularities of F(s). This ensures that the contour path is in the region of convergence. If all singularities are in the left half-plane, or F(s) is a smooth function on - ∞ < Re(s) < ∞ (i.e. no singularities), then γ can be set to zero and the above inverse integral formula above becomes identical to the inverse Fourier transform.

In practice, computing the complex integral can be done by using the Cauchy residue theorem.

It is named after Hjalmar Mellin, Joseph Fourier and Thomas John I'Anson Bromwich.

Post's inversion formula

An alternative formula for the inverse Laplace transform is given by Post's inversion formula.

See also

  • Inverse Fourier transform

References

  • Davies, B. J. (2002), Integral transforms and their applications (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-95314-4 
  • Manzhirov, A. V.; Polyanin, Andrei D. (1998), Handbook of integral equations, London: CRC Press, ISBN 978-0-8493-2876-3 

External links

This article incorporates material from Mellin's inverse formula on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Laplace transform — In mathematics, the Laplace transform is one of the best known and most widely used integral transforms. It is commonly used to produce an easily soluble algebraic equation from an ordinary differential equation. It has many important… …   Wikipedia

  • Laplace transform — Math. a map of a function, as a signal, defined esp. for positive real values, as time greater than zero, into another domain where the function is represented as a sum of exponentials. Cf. Fourier transform. [1940 45; after P. S. LAPLACE] * * *… …   Universalium

  • Laplace transform applied to differential equations — The use of Laplace transform makes it much easier to solve linear differential equations with given initial conditions.First consider the following relations:: mathcal{L}{f } = s mathcal{L}{f} f(0): mathcal{L}{f } = s^2 mathcal{L}{f} s f(0) f (0) …   Wikipedia

  • Two-sided Laplace transform — In mathematics, the two sided Laplace transform or bilateral Laplace transform is an integral transform closely related to the Fourier transform, the Mellin transform, and the ordinary or one sided Laplace transform. If f ( t ) is a real or… …   Wikipedia

  • Z-transform — In mathematics and signal processing, the Z transform converts a discrete time domain signal, which is a sequence of real or complex numbers, into a complex frequency domain representation. It is like a discrete equivalent of the Laplace… …   Wikipedia

  • Fourier transform — Fourier transforms Continuous Fourier transform Fourier series Discrete Fourier transform Discrete time Fourier transform Related transforms The Fourier transform is a mathematical operation that decomposes a function into its constituent… …   Wikipedia

  • Mellin transform — In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is often used …   Wikipedia

  • Integral transform — In mathematics, an integral transform is any transform T of the following form:: (Tf)(u) = int {t 1}^{t 2} K(t, u), f(t), dt.The input of this transform is a function f , and the output is another function Tf . An integral transform is a… …   Wikipedia

  • Weierstrass transform — In mathematics, the Weierstrass transform [Ahmed I. Zayed, Handbook of Function and Generalized Function Transformations , Chapter 18. CRC Press, 1996.] of a function f : R rarr; R is the function F defined by:F(x)=frac{1}{sqrt{4piint {… …   Wikipedia

  • Sumudu transform — In mathematics, the Sumudu transform, is an integral transform similar to the Laplace transform, introduced in the early 1990s by Gamage K. Watugala to solve differential equations and control engineering problems.Formal DefinitionThe Sumudu… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”